task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | The five tiles are randomly arranged in a row. Calculate the probability that the arrangement reads $XOXOX$. | \frac{1}{10} | 22 | 8 |
math | Given that the graph of the function $f(x)=\ln(x+m)$ is symmetric to the graph of $g(x)$ about the line $x+y=0$, and $g(0)+g(-\ln 2)=1$, find the value of $m$. | 2 | 57 | 1 |
math | The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \cos(a x)+4$ have for the same value of $a$? | 4014 | 75 | 4 |
math | Given the function $f(x)=e^{x}+2x-a$, where $a\in\mathbb{R}$, if there exists a point $(x_{0},y_{0})$ on the curve $y=\sin x$ such that $f(f(y_{0}))=y_{0}$, then the range of the real number $a$ is \_\_\_\_\_\_. | [-1+e^{-1},e+1] | 85 | 11 |
math | Find the prime number \( p \) if it is known that the number \( 13p + 1 \) is a perfect cube. | 211 | 30 | 3 |
math | In an equilateral triangle $∆ABC$ with side length $1$, let $\overrightarrow{BC}=2\overrightarrow{BD}$ and $\overrightarrow{CA}=3\overrightarrow{CE}$,
$(1)$ Find the projection of $\overrightarrow{AB}$ in the direction of $\overrightarrow{DA}$.
$(2)$ Find the value of $\overrightarrow{AD} \cdot \overrightarrow{BE}$. | -\frac{1}{4} | 90 | 7 |
math | Given triangle $\triangle ABC$ with side lengths $a=2$, $b=3$, and $c=4$, and altitude $h$ on side $BC$, calculate the value of $h$. | \frac{3\sqrt{15}}{4} | 42 | 13 |
math | A right triangle has side lengths $a, b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle. | 48+\sqrt{2016} | 45 | 10 |
math | Determine for which values of $x$ the expression \[\frac{x-20x^2+100x^3}{16 - 2x^3}\] is nonnegative. Answer as an interval. | [0, 2) | 48 | 6 |
math | Given the sequence $\{a_n\}$, where $a_1=t$ and $a_{n+1}= \frac{a_n}{2}+ \frac{2}{a_n}$, if $\{a_n\}$ is a monotonically decreasing sequence, determine the range of the real number $t$. | (2,+\infty) | 68 | 7 |
math | Given two statements p and q:
- p: There exists an $x$ in $\mathbb{R}$ such that $\cos 2x - \sin x + 2 \leq m$.
- q: The function $y = \left(\frac{1}{3}\right)^{2x^2 - mx + 2}$ is monotonically decreasing on the interval $[2, +\infty)$.
If either p or q is true (denoted by $p \lor q$) but not both (denoted by $p \land... | (-\infty, 0) \cup (8, +\infty) | 137 | 18 |
math | Given the universal set $U={4,m^{2}+2m-3,19}$, set $A={5}$, if the complement of $A$ relative to $U$ is ${|4m-3|,4}$, find the value of the real number $m$. | m=-4 | 63 | 3 |
math | Given that the function $f(x) = \sqrt{3}\sin\omega x - 2\sin^2\left(\frac{\omega x}{2}\right)$ ($\omega > 0$) has a minimum positive period of $3\pi$,
(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[-\pi, \frac{3\pi}{4}]$;
(II) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides o... | \frac{12 + 5\sqrt{3}}{26} | 212 | 17 |
math | Given that the prefix $ronna$ represents 27 zeros after the number, express $6ronna$ in the scientific notation $a\times 10^{b}$. | 6 \times 10^{27} | 38 | 10 |
math | Find the smallest real number $c$ such that \[|x_1| + |x_2| + \dots + |x_9| \geq c|M|\] whenever $x_1, x_2, \ldots, x_9$ are real numbers such that $x_1+x_2+\cdots+x_9 = 10$ and $M$ is the median of $x_1, x_2, \ldots, x_9$. | c = 9 | 105 | 4 |
math | Given a triangle \( \triangle ABC \) with area 1, and side length \( a \) opposite to angle \( A \), find the minimum value of \( a^2 + \frac{1}{\sin A} \). | 3 | 49 | 1 |
math | The quantity \(\tan 22.5^\circ\) can be expressed in the form
\[\tan 22.5^\circ = \sqrt{x} - \sqrt{y} + \sqrt{z} - w,\]
where \(x \ge y \ge z \ge w\) are positive integers. Find \(x + y + z + w.\) | 3 | 79 | 1 |
math | The diagram shows a regular dodecagon and a square, whose vertices are also vertices of the dodecagon. What is the value of the ratio of the area of the square to the area of the dodecagon? | 2:3 | 48 | 3 |
math | Given points $A(m,-2)$ and $B(3,n)$, if the line $AB$ is parallel to the $x$-axis and $AB=4$, find the value of $m+n$. | 5 \text{ or } -3 | 45 | 8 |
math | Given the function $f(x)= \begin{cases} kx+2, & x\geqslant 0 \\ \left( \frac {1}{2}\right)^{x}, & x < 0\end{cases}$, if the equation $f(f(x))- \frac {3}{2}=0$ has no solution in the set of real numbers, find the range of the real number $k$. | [0,+\infty) | 90 | 7 |
math | Find the coefficients \( m \) and \( n \) of the quadratic polynomial \( x^{2} + mx + n \), given that its remainders when divided by the binomials \( x - m \) and \( x - n \) are \( m \) and \( n \), respectively. | (0, 0), \left(\frac{1}{2}, 0\right), (1, -1) | 64 | 26 |
math | Find two numbers such that their sum is 24, their difference is 8, and their product is greater than 100. | 128 | 29 | 3 |
math | From points A and B, a motorcyclist and a cyclist set off towards each other at constant speeds simultaneously. Twenty minutes after the start, the motorcyclist was 2 km closer to point B than the midpoint of line AB. Thirty minutes after the start, the cyclist was 3 km closer to point B than the midpoint of line AB. A... | 24 | 88 | 2 |
math | Let $k\in \mathbb{N}$, $f(x)=\sin(\frac{2k+1}{3}\pi x)$. If for any real number $a$, the function $y=f\left(x\right)$ has the value $\frac{1}{3}$ at least $2$ times and at most $6$ times in the interval $\left[a,a+3\right]$, then determine the possible values of $k$. | 1 \text{ or } 2 | 96 | 8 |
math | Let $r$ , $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying
\[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \]
for all $n \ge 2023$ then the sum
\[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \]
is unbounded, i.e for all positive reals... | t = 1 + \frac{r}{s} | 168 | 12 |
math | What is the $33$ rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$? | 0 | 33 | 1 |
math | In a chess tournament each participant played exactly once against all others. The winner of each game received $1$ point, the loser $0$ points, and each scored $\frac{1}{2}$ point if the game was a draw. At the end of the tournament, it was found that exactly half of the points each player obtained came from playing a... | 24 | 110 | 2 |
math | Find the greatest common divisor of $7524$ and $16083$. | 1 | 20 | 1 |
math | What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | 10 | 44 | 2 |
math | In triangle $ABC$, $A(0,1)$, the equation of the line containing the altitude $CD$ from $AB$ is $x+2y-4=0$, and the equation of the line containing the median $BE$ from $AC$ is $2x+y-3=0$.
1. Find the equation of line $AB$;
2. Find the equation of line $BC$;
3. Find the area of $\triangle BDE$. | \frac {1}{10} | 99 | 8 |
math | How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities?
\[
\begin{aligned}
a^2 + b^2 &< 9, \\
a^2 + b^2 &< 8(a - 2), \\
a^2 + b^2 &< 8(b - 2).
\end{aligned}
\] | 2 | 77 | 1 |
math | The three vertices of $\triangle ABC$ are $A(4,0)$, $B(6,7)$, and $C(0,3)$. Find:<br/>
$(1)$ The equation of the median on side $AB$;<br/>
$(2)$ The equation of the altitude on side $BC$;<br/>
$(3)$ The equation of the perpendicular bisector on side $AC$. | 8x - 6y - 7 = 0 | 85 | 12 |
math | The number of integer solutions for the inequality \( |x| < 3 \pi \) is ( ). | 19 | 22 | 2 |
math | Given an ellipse C: $$\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with eccentricity $$\frac {\sqrt {2}}{2}$$, and vertices A($a$, 0), B(0, $b$), the distance from the center O to the line AB is $$\frac {2}{ \sqrt {3}}$$.
(1) Find the equation of ellipse C;
(2) Suppose there is a moving point P o... | 2 | 283 | 1 |
math | Given that the function $y=f(x)$ is increasing on $(0,3)$, and the function $y=f(x+3)$ is an even function, determine the relationship among $f(\frac{7}{2})$, $f(\frac{9}{2})$, and $f(2)$. | f(\frac{9}{2}) < f(2) < f(\frac{7}{2}) | 64 | 22 |
math | We need to color the three-element subsets of a seven-element set in such a way that if the intersection of two subsets is empty, then their colors are different. What is the minimum number of colors needed for this? | 3 | 44 | 1 |
math | How many common multiples do the numbers 180 and 300 have? Which of them is the smallest? If the smallest is determined, how to find the others? How many common divisors do these numbers have and what are they? | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 51 | 40 |
math | Given a function $f(x)$ and a real number $M$, if there exist $m$ and $n \in \mathbb{N}^+$ such that $f(m) + f(m+1) + f(m+2) + \ldots + f(m+n) = M$, then $(m, n)$ is called a "growth point" of the function $f(x)$ with respect to $M$. If $(1, 2)$ is a growth point of the function $f(x) = \cos(\frac{\pi}{2}x + \frac{\pi}... | 3 | 181 | 1 |
math | Given a geometric sequence $\{a_n\}$ with the first term $a_1$ and common ratio $q$, its general term $a_n$ is \_\_\_\_\_\_. | a_{1}q^{n-1} | 40 | 10 |
math | Given a sequence of natural numbers $\left\{x_{n}\right\}$ defined by:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \quad n=1,2,3,\cdots
$$
If an element of the sequence is 1000, what is the minimum possible value of $a+b$? | 10 | 91 | 2 |
math | A, B, and C are guessing a two-digit number.
A says: Its number of factors is even, and it is greater than 50.
B says: It is an odd number, and it is greater than 60.
C says: It is an even number, and it is greater than 70.
If each of them only spoke half the truth, what is this number? | 64 | 83 | 2 |
math | Consider a rectangular parallelepiped \( A_{1} B_{1} C_{1} D_{1}-A B C D \), with edge lengths \( A A_{1}=a, B_{1} A_{1}=b, A_{1} D_{1}=c \). Points \( M \), \( N \), \( P \), and \( Q \) are the midpoints of \( A_{1} B_{1} \), \( A_{1} D_{1} \), \( B C \), and \( C D \), respectively. Determine the distance between th... | \frac{1}{3} \sqrt{ a^2 + 4b^2 + 4c^2 } | 143 | 26 |
math | Given that $a, b \in \mathbb{R}$ and $a^2 + ab + b^2 = 3$, let the maximum and minimum values of $a^2 - ab + b^2$ be $M$ and $m$ respectively. Find the value of $M + m$. | 10 | 66 | 2 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the area of $\triangle ABC$ is ${S}_{\triangle ABC}=\frac{\sqrt{3}}{4}({a}^{2}+{b}^{2}-{c}^{2})$, then the range of $\frac{c}{a+b}$ is ______. | \left[\frac{1}{2}, 1\right) | 94 | 14 |
math | Taking the origin O as the pole and the positive half-axis of the x-axis as the polar axis, establish a polar coordinate system. The equation of line $l$ is $$\rho\sin(\theta- \frac {2\pi}{3})=- \sqrt {3}$$, and the polar equation of circle $C$ is $\rho=4\cos\theta+2\sin\theta$.
(1) Find the standard equations of lin... | \sqrt {19} | 130 | 6 |
math | If vectors $\overrightarrow{a}=(1,λ,1)$ and $\overrightarrow{b}=(2,-1,-2)$, and the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{{\sqrt{2}}}{6}$, calculate the value of λ. | -\sqrt{2} | 70 | 5 |
math | Given point P(3, -4), find the distance from point P to the chord AB formed by the two tangent lines drawn from point P to the circle C: $x^2+y^2=9$. | \frac{16}{5} | 44 | 8 |
math | The set of points $(x,y,z)$ that are equidistant to $(2,3,-4)$ and point $Q$ satisfy an equation of the form
\[8x - 6y + 32z = d.\] Find the point $Q$ and the value of $d$. | 151 | 63 | 3 |
math | Given $f(x)=x\ln x$, $g(x)=x^{3}+ax^{2}-x+2$
$(Ⅰ)$ Find the monotonic intervals of the function $f(x)$;
$(Ⅱ)$ For all $x\in (0,+\infty)$, if $2f(x)\leq {g'}(x)+2$ always holds, find the range of real number $a$. | a \geq -2 | 92 | 6 |
math | Let \( N \) be an even number that is not divisible by 10. What will be the tens digit of \( N^{20} \)? What will be the hundreds digit of \( N^{200} \)? | 3 | 49 | 1 |
math | For a real number \(x,\) find the maximum value of
\[
\frac{x^4}{x^8 + 2x^6 + 4x^4 + 8x^2 + 16}.
\] | \frac{1}{20} | 50 | 8 |
math | Given the function $$f(x)=2\sin^{2}\left( \frac {\pi}{4}+x\right)- \sqrt {3}\cos2x$$.
(1) Find the interval of monotonic decrease for the function $f(x)$;
(2) If the equation $f(x)=a$ has two distinct real solutions in the interval $$x\in\left[ \frac {\pi}{4}, \frac {\pi}{2}\right]$$, find the sum of these two solution... | (1, 4) | 164 | 6 |
math | Given the sets $M=\{x|x^2 = 2\}$, $N=\{x|ax=1\}$, determine the value of $a$ such that $N \subseteq M$. | 0, -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} | 44 | 23 |
math | Given $\cos (\alpha-\beta)= \frac {3}{5}$, $\sin \beta=- \frac {5}{13}$, and $\alpha \in (0, \frac {\pi}{2})$, $\beta \in (- \frac {\pi}{2},0)$, then $\sin \alpha=$ \_\_\_\_\_\_. | \frac {33}{65} | 74 | 9 |
math | For 8 **positive integers** $r$ and $n$ such that $1 \leqslant r \leqslant n$, find the arithmetic mean of the smallest number in all $r$-element subsets of the set $\{1,2, \cdots, n\}$, denoted as $f(r, n)$. | f(r, n) = \frac{n+1}{r+1} | 74 | 16 |
math | In how many different ways can 4 men and 3 women be placed into two groups of three people and one group of two people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable. | 72 | 53 | 2 |
math | Let $a$ and $b$ be real numbers. Consider the following five statements:
1. $\frac{1}{a} > \frac{1}{b}$
2. $a^2 < b^2$
3. $a > b$
4. $a > 0$
5. $b > 0$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? | 3 | 94 | 1 |
math | Two cars, Car A and Car B, simultaneously depart from locations A and B respectively, traveling towards each other at a constant speed. They meet at a point 60 kilometers from location A. After meeting, they continue to their respective destinations (Car A reaches B and Car B reaches A) and then immediately return. The... | 130 | 90 | 3 |
math | Let $x, y, z$ be real numbers such that
\begin{align*}
y+z & = 24, \\
z+x & = 26, \\
x+y & = 28.
\end{align*}
Find $\sqrt{xyz(x+y+z)}$. | \sqrt{83655} | 63 | 9 |
math | Given \\(|a| = \sqrt{2}\\), \\(|b| = 1\\), and \\(a \perp (a + 2b)\\), find the angle between vector \\(a\\) and vector \\(b\\). | \frac{3\pi}{4} | 54 | 9 |
math | In the rectangular coordinate system $xOy$, the equation of curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of line $l$ is given by $\rho \sin(\theta - \frac{\pi}... | \sqrt{7} + 2 | 177 | 8 |
math | Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \] | (6, 2, 10) | 52 | 11 |
math | Let the first term of a geometric sequence be $\frac{5}{6}$, and let the second term be $25$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by ten million? | 8 | 51 | 1 |
math | Given positive integers \(a, b, c, x, y, z\) that satisfy the conditions
$$
a \geq b \geq c \geq 1, \quad x \geq y \geq z \geq 1,
$$
and
$$
\begin{cases}
2a + b + 4c = 4xyz, \\
2x + y + 4z = 4abc,
\end{cases}
$$
determine the number of six-tuples \((a, b, c, x, y, z)\) that satisfy the given conditions. | 0 | 129 | 1 |
math | Given the hyperbola $\frac{x^{2}}{4}-y^{2}=1$, and $F_{1}$, $F_{2}$ are its two foci, find the value of $|\overrightarrow{PF_{1}}|\cdot|\overrightarrow{PF_{2}}|$. | 2 | 63 | 1 |
math | Determine the value of the following product with a short calculation:
$$
\frac{6 \cdot 27^{12}+2 \cdot 81^{9}}{8000000^{2}} \cdot \frac{80 \cdot 32^{3} \cdot 125^{4}}{9^{19}-729^{6}}
$$ | 10 | 86 | 2 |
math | In a three-dimensional space, we have three mutually perpendicular planes: $\alpha$, $\beta$, and $r$. Let there be a point $A$ on plane $\alpha$. Point $A$ is at a distance of $1$ from both planes $\beta$ and $r$. Let $P$ be a variable point on plane $\alpha$ such that the distance from $P$ to plane $\beta$ is $\sqrt{... | 0 | 123 | 1 |
math | Calculate: $(-1)^{3} + 4 \times (-2) - 3 \div (-3)$. | -8 | 26 | 2 |
math | We have a standard deck of 52 cards, divided equally into 4 suits of 13 cards each. Determine the probability that a randomly chosen 5-card poker hand is a flush (all cards of the same suit). | \frac{33}{16660} | 47 | 12 |
math | Standa assembled 7 identical structures, each made of 8 identical gray cubes with an edge length of $1 \mathrm{~cm}$ as shown in the picture.
Then, he immersed them all in white paint and subsequently disassembled each structure back into the original 8 parts, which now had some faces gray and others white. He also ad... | 12 \text{ cm}^2 | 128 | 9 |
math | Given two non-empty sets $A$ and $B$ satisfying $A \cup B = \{1, 2, 3\}$, calculate the number of ordered pairs $(A, B)$. | 25 | 43 | 2 |
math | Given the plane vectors $\overrightarrow{a}, \overrightarrow{b}$ with magnitudes $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=\frac{1}{3}|\overrightarrow{a}|$, as well as the magnitude of the difference between them $|\overrightarrow{a}-\frac{1}{2}\overrightarrow{b}|=\frac{\sqrt{43}}{3}$, determine the angle betw... | \frac{2\pi}{3} | 110 | 9 |
math | Given that the complex number $z$ satisfies $z(2-i) = 11+7i$ (where $i$ is the imaginary unit), determine the value of $z$. | 3+5i | 40 | 4 |
math | Given the function $f(x)=\sqrt{x^2+2x+1}-|x-m|$ has a maximum value of 4 (where $m>0$).
(1) Find the value of $m$;
(2) If $a^2+b^2+c^2=m$, find the minimum value of $\frac{1}{a^2}+\frac{1}{b^2}+\frac{4}{c^2}$. | \frac{16}{3} | 99 | 8 |
math | Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to a larger circle $D$. Circles $B$ and $C$ are congruent. Suppose circle $A$ has radius 2 and the center of circle $D$ is on circle $A$, and the radius of circle $D$ is twice the diameter of $A$. What is the radius of circle $B$? | 2 | 90 | 1 |
math | Mark borrows $20 from Emily at a simple interest rate of $10%$ every day. On the third day, he borrows an additional $15. What is the least integer number of days after which Mark will have to pay her back at least twice the total amount he borrowed? | 12 \text{ days} | 62 | 7 |
math | Point A(m, y1) and B(m+1, y2) are both on the graph of the quadratic function y=-(x-1)^2+n. If y1 > y2, determine the range of values for m. | m > \frac{1}{2} | 50 | 9 |
math | In triangle \( \triangle ABC \) inscribed in the unit circle, the internal angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circle again at points \( A_1 \), \( B_1 \), and \( C_1 \) respectively. Find the value of \( \frac{A A_1 \cos \frac{A}{2} + B B_1 \cos \frac{B}{2} + C C_1 \cos \frac{C}{2}}{\... | 2 | 125 | 1 |
math | Given a positive sequence $n$ with the sum of the first $n$ terms denoted as $S_n$, and $a_1=1$, $a_{n+1}^2=S_{n+1}+S_n$.
- $(1)$ Find the general formula for the sequence $\{a_n\}$.
- $(2)$ Let $b_n=a_{2n-1}\cdot 2^{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=(2n-3)\cdot 2^{n+1}+6 | 120 | 18 |
math | If point $P(\cos \alpha, \sin \alpha)$ lies on the line $y=-2x$, then the value of $\sin 2\alpha + 2\cos 2\alpha$ is ____.
A) $- \frac {14}{5}$
B) $- \frac {7}{5}$
C) $-2$
D) $\frac {4}{5}$ | -2 | 87 | 2 |
math | In base \( R_1 \), the fractional expansion of \( F_1 \) is \( 0.373737 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.737373 \cdots \). In base \( R_2 \), the fractional expansion of \( F_1 \) is \( 0.252525 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.525252 \cdots \). What is the sum of... | 19 | 146 | 2 |
math | In the trapezoid $ABCD$, the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the diagonal $AC$, $N$ the midpoint of the diagonal $BD$, and $P$ the midpoint of the side $AB$. Given that $AB = 15 \text{ cm}$, $CD = 24 \text{ cm}$, and the height of the trapezoid is $h = 14 \text{ cm}$:
a) Calculate the leng... | 15.75 \text{ cm}^2 | 130 | 12 |
math | Given the coordinates of points A(0,30), B(20,10), and O(0,0), determine the number of lattice points inside triangle ABO. | 271 | 38 | 3 |
math | A chemical plant introduced an advanced production line to produce a certain chemical product. The relationship between the total production cost $y$ (in ten thousand yuan) and the annual output $x$ (in tons) can be approximately expressed as $y=\frac{x^2}{5}-48x+8000$. It is known that the maximum annual output of thi... | 1660 | 152 | 4 |
math | Given the hyperbola $\Gamma$: $x^{2}- \frac{y^{2}}{8}=1$ with left and right foci $F\_1$ and $F\_2$, respectively. Let $A$ be the left vertex of the hyperbola $\Gamma$, and the line $l$ passes through the right focus $F\_2$ intersecting the hyperbola $\Gamma$ at points $M$ and $N$. If the slopes of $AM$ and $AN$ are $k... | y=-8(x-3) | 144 | 7 |
math | Given the functions $f(x)=2\cos^{2}(x+\frac{\pi }{12})$ and $g(x)=3+2\sin x\cos x$.
(I) Find the equation of the axis of symmetry for the graph of the function $y=f(x)$.
(II) Find the smallest positive period and the range of the function $h(x)=f(x)+g(x)$. | [3,5] | 87 | 5 |
math | The distance from the center of the circle $x^{2}+y^{2}-2x=0$ to the line $2x+y-1=0$ is $\frac{\sqrt{5}}{5}$. | \frac{\sqrt{5}}{5} | 47 | 10 |
math | What is the smallest positive integer \( n \) for which \( 3^{2n} - 1 \) is divisible by \( 2^{2010} \)? | 2^{2007} | 38 | 7 |
math | Given \(\frac{1+\sin x}{\cos x}=\frac{22}{7}\), and \(\frac{1+\cos x}{\sin x}=\frac{m}{n}\) (where \(\frac{m}{n}\) is in simplest form). Find \(m+n\). | 44 | 67 | 2 |
math | What is the nearest integer to $(3+\sqrt{2})^6$? | 7414 | 17 | 4 |
math | In $\triangle ABC$, $AB= \sqrt {2}$, $BC=1$, $\cos C= \dfrac {3}{4}$.
(I) Find the value of $\sin A$;
(II) Find the value of $\overrightarrow{BC}\cdot \overrightarrow{CA}$. | \overrightarrow{BC}\cdot \overrightarrow{CA} = -\dfrac{3}{2} | 64 | 23 |
math | Given the expression $(x^2 + y^2)^{-1} + (x^{-2} + y^{-2})$, simplify the expression. | (x^2 + y^2)x^{-2}y^{-2} | 31 | 15 |
math | Find the remainder when the polynomial $(2x)^{500}$ is divided by the polynomial $(x^2 + 1)(x - 1).$ | 2^{500} x^2 | 34 | 9 |
math | Two congruent isosceles right triangles are placed so that they overlap partly and their hypotenuses coincide, each with a hypotenuse of 10. Find the area of the region where the triangles overlap. | 12.5 | 45 | 4 |
math | The output of a factory increased 4 times over four years. By what average percentage did the output increase each year compared to the previous year? | 41.4\% | 29 | 6 |
math | Given that $F$ is the right focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, a perpendicular line is drawn from point $F$ to one of the asymptotes of $C$, with the foot of the perpendicular being $M$, and intersecting the other asymptote at point $N$. If $3\overrightarrow{MF} = \... | \frac{\sqrt{6}}{2} | 129 | 10 |
math | Given $\sin x = \frac{3}{5}$, with $x \in \left( \frac{\pi}{2}, \pi \right)$, find the values of $\cos 2x$ and $\tan\left( x + \frac{\pi}{4} \right)$. | \frac{1}{7} | 63 | 7 |
math | If the function \( f(x) = \log_{a}\left(4x + \frac{a}{x}\right) \) is increasing on the interval \([1, 2]\), what is the range of values for \( a \)? | (1, 4] | 53 | 6 |
math | The spinner now has six congruent sectors, labeled 1 through 6. Jane and her brother each spin the spinner once. If the non-negative difference of their spin results is at least 2 but less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? | \frac{7}{18} | 63 | 8 |
math | A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box. Winning is defined as drawing two balls of... | \frac{6}{7} | 173 | 7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.